A BAYESIAN-APPROACH TO NONLINEAR INVERSION
A BAYESIAN-APPROACH TO NONLINEAR INVERSION
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DOI：
10.1029/jb090ib01p00581
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发表时间：
1985-01-01
期刊：
影响因子：
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通讯作者：
MATSUURA, M
中科院分区：
文献类型：
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作者：
JACKSON, DD;MATSUURA, M
迭代线性反演理论常用于求解非线性反演问题。此外，线性反演理论提供了方便的误差估计和其他解释措施。但是这些解释性措施对非线性问题有效吗？我们解决这个问题的联合概率密度函数（PDF）的估计参数。简而言之，如果观测值是关于最佳估计的合理（例如，95%）置信区域内的参数的线性函数，如果最佳估计是唯一的，则线性反演理论将是有效的。我们使用贝叶斯规则来说明先验信息如何提高最优估计的唯一性，同时稳定此估计的迭代搜索。我们还制定了定量标准的相对重要性的先验和观测数据和非线性的影响。我们的方法可以处理任何形式的pdf的观测数据和先验信息。当观测数据和先验数据都是高斯分布时，计算要容易得多（大约与马夸特方法所需的相同）。我们给出了一些简单的单参数和双参数非线性反问题的计算。这些例子表明，渐近统计（那些基于线性理论）在某些情况下可能是严重错误的。在其他情况下，精确的观察、先验信息或两者的组合可以有效地线性化原本非线性的问题。
Iterated linear inversion theory can often solve nonlinear inverse problems. Also, linear inversion theory provides convenient error estimates and other interpretive measures. But are these interpretive measures valid for nonlinear problems? We address this question in terms of the joint probability density function (pdf) of the estimated parameters. Briefly, linear inversion theory will be valid if the observations are linear functions of the parameters within a reasonable (say, 95%) confidence region about the optimal estimate, if the optimal estimate is unique. We use Bayes' rule to show how prior information can improve the uniqueness of the optimal estimate, while stabilizing the iterative search for this estimate. We also develop quantitative criteria for the relative importance of prior and observational data and for the effects of nonlinearity. Our method can handle any form of pdf for observational data and prior information. The calculations are much easier (about the same as required for the Marquardt method) when both observational and prior data are Gaussian. We present calculations for some simple one and two‐parameter nonlinear inverse problems. These examples show that the asymptotic statistics (those based on the linear theory) may in some cases be grossly erroneous. In other cases, accurate observations, prior information, or a combination of the two may effectively linearize an otherwise nonlinear problem.